Abstract: Among specialists prevails the primitive view, according to Prof. G.S. Ivanov, on descriptive geometry only as on a \"grammar of a technical language\", as it characterized V.I. Kurdyumov in the XIX Century. If in the century before last his definition was actual, although many contemporaries had a different opinion, then a century and a half later this definition became outdated, especially since have been revealed the close relationships of descriptive geometry with related sections: analytical, parametric, differential geometry, etc., and descriptive geometry became an applied mathematical science. In this paper it has been shown that an image is obtained as a result of display (projection). In this connection, according to prof. N.A. Sobolev, \"All visual images – documentary, geometrographic, and creative ones – are formed on the projection principle\". In other words, they belong, in essence, to descriptive geometry. Thus, all made by hand creative images – drawings, paintings, sculptures – can be attributed with great confidence to descriptive geometry as a theory of images. These creative images, of course, have a non-obvious projection origin, nevertheless, according to Prof. N.A. Sobolev, \"They, including the most abstract fantasies, are essentially the projection ones\". Further in the paper it has been shown which disciplines apply some or other of graphic models, and has been considered a number of drawings belonging to different textbooks, in which graphic models are present. Thus, clearly, and also referring to the authorities in the area of images and descriptive geometry, it has been proved that each of the mentioned textbooks has a direct or indirect connection with descriptive geometry, and descriptive geometry itself is present in all textbooks, at least, in the technical and medical ones.

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Topics: Descriptive geometry (64%)

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Abstract: Physical spline is a resilient element whose cross-sectional dimensions are very small compared to its axis’s length and radius of curvature. Such a resilient element, passing through given points, acquires a "nature-like" form, having a minimum energy of internal stresses, and, as a consequence, a minimum of average curvature. For example, a flexible metal ruler, previously used to construct smooth curves passing through given coplanar points, can be considered as a physical spline. The theoretical search for the equation of physical spline’s axis is a complex mathematical problem with no elementary solution. However, the form of a physical spline passing through given points can be obtained experimentally without much difficulty. In this paper polynomial and parametric methods for approximation of experimentally produced physical spline with large deflections are considered. As known, in the case of small deflections it is possible to obtain a good approximation to a real elastic line by a set of cubic polynomials ("cubic spline"). But as deflections increase, the polynomial model begins to differ markedly from the experimental physical spline, that limits the application of polynomial approximation. High precision approximation of an elastic line with large deflections is achieved by using a parameterized description based on Ferguson or Bezier curves. At the same time, not only the basic points, but also the tangents to the elastic line of the real physical spline should be given as boundary conditions. In such a case it has been shown that standard cubic Bezier curves have a significant computational advantage over Ferguson ones. Examples for modelling of physical splines with free and clamped ends have been considered. For a free spline an error of parametric approximation is equal to 0.4 %. For a spline with clamped ends an error of less than 1.5 % has been obtained. The calculations have been performed with SMath Studio computer graphics system.

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Topics: Spline (mathematics) (65%), Tangent (52%), Curvature (52%) ... show more

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Topics: Descriptive geometry (57.99%), Computer graphics (56.99%), Base (topology) (52%)

27 Citations

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Андрей Пантелеев, Andrey Panteleev, Инна Ковтуненко, Inna Victorovna Kovtunenko^{1}•Institutions (1)

08 Jul 2019-

Abstract: The textbook contains theoretical information on morphemics and word formation of the modern Russian language; Glossary of terms, plans of practical classes, tasks and exercises for them; tasks for self-control, options for tests and tests; schemes and samples of analysis of language units, a list of scientific and educational literature; questions for the exam. Prepared in accordance with the Federal state educational standard of higher education in directions of preparation "Pedagogical education", "Philology", in accordance with the approximate program of the course "Modern Russian language" and is intended for students enrolled in the profile "Russian language and literature, Russian language and foreign language", "national Philology", for students of the specialist degree, students majoring in "Russian language and literature", master of Philology, as well as for foreign students studying Russian language.

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Topics: Word formation (70%)

17 Citations

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Abstract: In this paper are considered historically the first (the 60’s of the 20th century) computational methods for algebraic cubic curves constructing. The analysis of a general cubic curve equation r(t)=a3t3+a2t2+a1t+a0 \nhas been carried out. As an example has been considered the simplest cubic curve r(t)=it3+jt2+kt. \nBased on the general cubic curve equation have been obtained equations of a cubic curve passing through two predetermined points and having predetermined tangents at these points. \nThe equations have been presented both in Ferguson and Bézier forms. It has been shown that the cubic curve vector equation (for example, the standard equation of a Bezier curve) can be represented in a point form. Have been considered examples for constructing segments of cubic curves meeting the given boundary conditions. \nThe generalized cubic curve equation, containing weight coefficients, has been obtained by the method of exit into four-dimensional space. Has been considered a vector parametric equation of a conical section, passing through two given points and touching predetermined straight lines at these points. The conical section is considered as a special case of a cubic curve. \nCurvature can be specified as an additional boundary condition. Has been considered the possibility for constructing a cubic curve with fixed positions of contacting planes at end points and given radii of curvature. Has been proposed an algorithm for constructing a plane cubic curve with a given curvature at the end points. \nHave been considered algorithms for constructing smooth compound Ferguson-Bezier curves. Smoothness conditions are imposed on a compound curve: 1) at any of its points, the curve must have a tangent (no fractures are allowed), 2) the curvature vector must be changed continuously from point to point (no discontinuous jump in the curvature vector is allowed neither in modulus no in direction). Have been proposed examples for constructing compound Ferguson-Bézier curves. \nHas been performed comparison of polynomial cubic spline with compound parametrically defined curves. Have been given examples for constructing cubic splines with fastened and free ends. \nThe paper is for educational purposes, and intended for in-depth study of computer graphics basics.

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6 Citations